In Dormand-Prince original paper for Runge Kutta 4(5) formula (link: https://www.sciencedirect.com/science/article/pii/0771050X80900133?via%3Dihub), at the beginning, it is noted that the equation
y' = f(t, y(t))
by taking t'=1, without the loss of generality, simplifies to
y' = f(y(t))
And then it procedes with taylor series expansion of y by using simplified equation. Now the question is, Is the taylor series expansion of y by using any of the above equations are equivalent? That is, Is taking
y'' = df(t,y(t))/dt + df(t,y(t))/dy * f
is equal to
y'' = df(y(t))/dy * f.
If not how we do not lose generality?
Yes, it is the same, you just extend the system of ODE and thus the Jacobi matrix and other derivative tensors by some trivial constant columns by adding the time as a state component.
If the paper did state it the way you cited it, then they mean that instead of considering the class of general ODE $y'=f(t,y)$, it is sufficient (and shorter in terms of notation, which is why they did that) to consider the class of autonomous ODE $y'=\tilde f(\tilde y)$, where $\tilde y=(t,y)=(t,y_1,...,y_n)$ and $\tilde f=(1,f)=(1,f_1,...,f_n)$. In the citation this is shortly indicated by the hint that $t'=1$ is added to the system as additional equation, with $t$ implied as additional state component.
Using different letters instead of modifiers, set $u=\pmatrix{t\\y}$ and $u'=F(u)=\pmatrix{1\\f(t,y)}$ according to $t'=\frac{dt}{dt}=1$. Then for the second derivative of $u$ you get $$ u''=F'(u)u'=F'(u)F(u)=\pmatrix{0&0\\\partial_tf(t,y)&\partial_yf(t,y)}\pmatrix{1\\f(t,y)} =\pmatrix{0\\\partial_tf(t,y)+\partial_yf(t,y)f(t,y)}. $$ This has the expected entries in the "non-trivial" positions.